Braids and the Configuration Space of Points

One can view a n string braid as a loop in the configuration space of n points in the plane. I made a quickie screencast to illustrate this.

Actually I was trying to see if I could get SketchUp to animate the moving points seen in the first half of the screencast. In the end I did it manually with the assistance of section planes. The second half however is done automatically. Get the model and click on the beginsweep/endsweep buttons after opening it in SketchUp.

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3 Responses to “Braids and the Configuration Space of Points”

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bye!

I had a question regarding the configuration space of n particle systems.
One of the defining properties of braids require that no two strings pass through the same point. If instead it is allowed, then the strings would practically be independent of each other. Because almost all interesting braids will be homotopically equivalent. So if I understand correctly, there won’t be anything non trivial if we let the strings pass/cut through each other

Now from the physics point of view, time evolution of an n-particle quantum system in the configuration manifold corresponds to a braid of n strings. But the reason that this is a correct model for n-particle system is that we do not let points of the form ( x_1,x_1,x_2….x_(n-1) ), where two or more coordinates are the same, be in the configuration space of n particles.
I find it hard to justify this requirement on the configuration space of n particles. Leinaas and Myrheim, in the paper ‘On the theory of identical particles’ justifies this elimination of repeated coordinates (they call such points singular, I don’t know why) by saying that they have measure 0 (I think I can see that but this convenience does not justify the assumption).

Do you know if their is any real physical motivation to remove those points from the config space?

An instance of repeated coordinates means you’re at a time where you’ve got (at least) two particles in the same position. Assuming two different things don’t occupy the same space, I reckon it’s reasonable to not include such points in the config space. Points at which a space fails to be locally a manifold (like where two strands of a ‘braid’ actually meet) are often called singular.